NRL Report 7349 - DTIC · 2018-11-08 · NRL Report 7349 A310310B/652A/A R02101-002 c" (formerly A310310B/652A/1 elf AOAOI lC.E)A1 R008-01-020) OlI~t~1lV~l ~ 9b. OTHER thie repot,)REPORT

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Naval Research Laboratory Unclassified2b. GROUP

1 3 REPORT TITLE

ON THE STATISTICS OF SEA CLUTTER

4 OESCRIPTIVE NOTES (Type of report and Inclusive dates)

A final report on one phase of a continuing NRL problem.__ b , Au I uo-tat (First name, middle initial, lost naMe)

G.R. Valenzuela and M.B. Laing

0 REPORT DATE 78. TOTAL NO. CoF PAGES 17b. NO. OF RPFS

December 30, 1971 38 231,. CONTRACT OR GRANT NO. 9A. ORIOINATOR'S REPORT NUMBERISt

NRL Problem R02-37b. PROJECT NO. NRL Report 7349

A310310B/652A/A R02101-002elf AOAOI lC.E)A1 OlI~t~1lV~l ~ 9b. OTHER REPORT NO(S) (Aniy other rnucbers, that may be eassmnedc" (formerly A310310B/652A/1 R008-01-020) thie repot,)

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Approved for public release; distribution unlimited.

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Department of the Navy (Naval Air SystemsCommand), Washington, D.C. 20360

I). AhSTRACT

A model for the statistics of sea clutter has been developed from scattering theory and the compositesurface-spattering model. The model postulates that sea clutter is exponentially (Rayleigh envelope) dis-tributed for glassy seas and should tend toward the lognormal distribution (in particular for horizontalpolarization) with increasing roughness. The lognormality of sea clutter arises from the tilting of theslightly rough "patches4 by the large-scale roughness (undulating surface).

An empirical identification of the statistics of sea clutter taken with the Four-Frequency Radarsystem shows that in general the distribution of sea clutter is intermediate between the exponential(Rayleigh envelope) and the lognormal distribution. However, for calm seas and small sample sizes (lessthan about 200 independent samples) the distribution of sea clutter may be approximated by eitherthe exponential or the lognormal distribution.

The first five central momenta of sea clutter (in decibels) have been calculated for moderte andrough sea conditions.

D ORM (PAGE I )S00., , 0, 36S/N 0101. 807.6600t1t-d c•'astnc~te .....

Security Ciassafication

LINK A LINK U LINK CKEY WOROS

ROLE WT ROLE WT ROLE WT

Sea clutterStatistical modelScattering theory

Composite surfaceProbability distribution functionsProbability density functionsLognormal clutterKolmogorov-Smirnov testIndependent SamplesMoments

DD Novell1473 1(BACK) 36(PAGE 2) Security Clashification

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TABLE OF CONTENTS

Pap

Abbt ............................................................ iiProblem Sttru ....................................................... AAuthorization . ....................................................... ii

INTRODUCTION .................................................... I

THEORETICAL CONSIDERATIONS ..................................... 2llbe Radw e• Pmfrim ...k.m..................................... 2Eis'-1 C o u .- Scattaifg from Rouda S&rwm ........................ .3

STATWIICAL MODEL OF BRA CLATr]a ............................ .3

ID'IT-FICATION OF THE STATISTIC OF SEA CLUTrER OHNrA2WIH THE NR.L-4FR SYSEM -....................................... 15

T e Db *ub Fmetiu .......................................... 15Th1e Cas . Wamwb ....... ....................................... 17

DISCUSSION OF REXULTS AN)D CONCLUSIONS ............... ..... 26

ACKNOWLFDGMENTS ............................................. 33

REFER•CES........ .............. 33

iI

~.

ABSTRACT

A model for the statistics of sea clutter has been developed from scattering theory andthe composite surface-scattering model. The model postulates that sea clutter s exponentially(Rayleigh envelope) distributed for gla -7 -9s -r.d should tend toward the lognormal distribu-

tion (in particular for horizontal polarizauton) with increasitig roughness. The lognorma~ityof sea clutter arises from the tilting of the slightly rough "patches" by the large-scale rough-ness (undulating surface).

An empirical identification of the statistics of sea clutter taken with the Four-FrequencyRadar system shows that in general the distribution of sea clutter is intermediate betweenthe exponential (Rayleigh envelope) and the lognormal distribution. However, for calm seasand small sample sizes (less than about 200 independent samples) the distribution of seaclutter mmy be approxi... t• d by citcr the exponential or the lognornrial distzibutLkn.

The first five central moments of sea clutter (in decibels) have been calculated formoderate and rough sea conditions.

PROBLEM STATUS

This is a final report on one phase of the problem; work on other phases is continuing.

AUTHORIZATION

NRL Problem R02-37Project A310310B/652A/A R02101-002

(formerly A310310B/652A/1 R008-01-020)

Manuscript submitted September 10, 1971.

i

ON THE STATISTICS OF SEA CLUTTER

INTRODUCTION

Radar detection, which is a binary detection problem (i.e., the selection of one outcomefrom two events), has developed quite rapidly since the early 1940's and its progress can be

followed in Middleton (1). However, modern radar detection theory is not considered tohave started until the Marcum (2) and Swerling (3) contributions, and presently it hasreached a high level of maturity. Nevertheless, most formal developments still only treat

explicitly the detection of signals (deterministic or random) strictly in Gaussian noise (4).

Unfortunately, many of the geophysical noises encountered in practice are non-Gaussian(e.g., atmospheric disturbances, terrain, and sea clutter), and to implement realistic detectionschemes the true statistics of these geophysical noises must be identified a priori.

Recently it has been suggested that the statistics of sea clutter, in particular for high.resolution radars and toward grazing incidence, cannot be expressed as a Rayleigh distribu-tion (exponential in power) but can be approximated by other distributions, among them thelognormal distribution (5).

The main purpose of this investigation is to identify the statistical properties of seaclutter, both by electromagnetic scattering theory via the composite surface-scatteringmodel (6, 7) and empirically by statistical analysis of sea clutter taken -with the Four-Frequency Radar (4FR) system (8), which transmits at 428 MHz (P band or UHF), 1228MHz (L band), 4455 MHz (C band), and 8910 MHz (X band). The composite surface-scattering model already has been quite successful in predicting the mean value of seaclutter (9-11) and explaining the width of the doppler spectrum of radar sea echo (12). Thepotential of this scattering model to derive other statistical information of sea clutter is tobe explored here.

Anticipating some of the results to be obtained later, it is possible to say that the tiltingof the slightly rough "patches" by the large-scale roughness (i.e., the undulating surface) isthe mechanism that generates the non-Rayleigh statistics observed in sea clutter, in particularfor horizontal polarization. Also in the empirical identification of sea clutter taken with the4FR system Ly the Kolmogorov-Smirnov test on the cumulative distribution and by thecomputation of the first five central moments, the distribution of sea clutter (for a radarpulsewidth of 0.5 psecond) is found to be intermediate between the expohential (Rayleighenvelope) and the lognormal distribution, and these distributions may serve as the limitingdistributions of sea clutter.

1!

2 G.R. VALENZUELA AND M.B. LAING

THEORETICAL CONSIDERATIONS

The Radar Detection Problem

In binary detection problems the Bayes criterion is used (1, 4) when the a priori prob-abilities and cost functions of making a decision are known (i.e., the risk in making the wrongdecisions is minimized). The implementation of the Bayes criterion in detection leads to alikelihood ratio test

p(ViHo > X, ifH, is true, (la)

and

A(V) =P(Vh < X, if Ho is true, (ib)p( V/Ho)

wherep(V/H1 ) andp(V/Ho) are the conditional probability densities given that event H1 istrue (signal plus noise are present) and given that event H0 is true (noise alone is present),respectively. The threshold level X is a function of the cost functions and the a priori prob-abilities of the sources.

Since, in practice, the cost functions and the a priori probabilities ore not known, othercriteria must be considered. In radar detection, the Neyman-Pearson criterion is most fre-quently used, the probability of false alarm PF is specified, and the probability of detectionPD is maximized.

The threshold level under these constraints is obtained from

PF f p(VIH°) dV, (2)

and the required signal-to-nois- ratio required for detection then is obtrirnd from

PD- j p(VIH1 )dV. (3)

The implementation of the Neyman-Pearson criterion also leads to a likelihood ratio testsimilar to Ea. (1) with the threshold level now being determined from Eq. (2).

Thus, for radar systems operating in the sea environment the conditional probabilitydensity of clutter must be known in order to have realistic r-limates of the probability offalse alarm as a function of radar and sea parameters. Othet ci iteria for detection alsoExict (1, 4), but thcy will' riot be discussed here. I

1

NRL REPORT 7349 3

Integration of several radar pulses should improve the performance of detec ion, theamount of improvement depending on the correlation properties of clutter alone in relationto the correlation properties of signal plus clutter.

Electromagnetic Scattering from Rough Surfies

Consider the scattering properties of a statistical rough surface, with surface currentsKe 99 n X H and Km -- n X E due to a plane wave incident on the surface (Fig. 1). Thescattered fields may be obtained by means of the following integral equation formulation(13, 14), where the time dependence is iaker, to be eiw t and the unit normal vector n ispointing toward free space:

E.(r) - Ids'K,.(r') ' XVG(rr'P- Ti° Y fV K6(r')XV'G(r~r') (4)

.• ' ' 1 .. fds' K,(r') XV'G(r'r'), (4)a af

H,(r) -- ds'K.(r')XV'G(r~r')+ VX da' K,( (r) X 'G(r,r)), (5

where w is the electromagnetic frequency in radians, p0 is the magnetic permeability, k isthe propagation constant of a plane wave in free space, G(r,r') is the Green's function in frF-space, and r' is the radiu• vwor 'or a point on the rough surface.

SCATTEREDWAVES

/ /I/ /

INC IDENTWAVE k2

pit I ./Ste.Aaa goonmry

I //p

4 G.R. VALENZUELA AND M.B LAING

The integral equations (4) and (5) can be simplified for the "far field" (i.e., all scatteredrays arriving in a given direction must be propagating parallel to each other). In this case,the following approximations apply

V'G(r,r') - ' f41r-r' ikG(r~r')n 2 (6)

G(R,r') - -ikRe" (7"

VX KW(r') X V'G(r,r')-V XV'G(r,r') X K,(r')oz-k 2 n2 X (n2 X K,)G(r,r'), (8)

where k2 is the propagation vector of the scattered wave, R is the distance from the obser-vation point to the origin of Lhe coordinates system, and n2 - k 2/k.

Using the approximations given in Eqs. (4) and (5), the following equations result:

E8(R)IJ( {fd' n2 X [Km(r')+Zon 2 X K(r')]Jeik 2f (9)

H*(,R) fd'NX [ K,')- Y~n X K.(')]eak 2 ~ (10)

wheref

-ike-ikR

K4wR ' Z0 - V e 120M , and Y0 - 1/IZo.

Ac•.ýrdingly, the statistics of the scattered fields depend only on the statistics of the surfacecurrents and the statistic properties of the rough surface.

Letting

L=fsK"60 , 2M-fds' K,(r)e4k 2", (12)

the magnitude square of the fields (which is proportional to power) is given by

IE"PIS p12 {IL'(p X n2 )12 + Z8IM'p 2 + 2Zo0 e(M-p)(L*-(p X n2) ] (13)

and

XEn2 pi12 k 1•L'p12 +Z2 IM'(n" X p)l 2 2Z°Ae(L*'p)IM'(n 2 X p)] ,(14)

where L* is the complex conjugate vector of L and p is a polarization unit vector such thatp'n 2 - 0.

NRL REPORT 7349 5

For a rough surface of impedance Z., the magnetic surface current Km (r') cAn be ob-tained from the electric surface current K, (r') by

Km(r') = -Z.n(r') X K,(r'). (15)

The surface currents K,, and Km are determined from the boundary value problem ofthe air/rough surface interface from the incident electromagnetic wave on the surface. Whenthe scattering surface is statistically rough, the boundary value problem cannot be solved inclosed form, and accordingly appropriate approximations must be used for th? surfacecurrents.

Fortunately, for rough surfaces like the sea, which has a two-dimensional spectrumW(K) of the form K- 4 (K is the wavenumber of the surface roughness or waves), a slightlyrough local boundary condition may be used. This fundamentally separates the sea surfaceinto luage-scale roughnesses (undulating surface) and small-scale roughneses (Bragg resonantscatterers). According to the local-slightly rough boundary condition, the surface fields onthe undulating surface (ie., Or')) may be expressed as a function of the small-scale rough-ness (i.e., E(r')), |

H(r') - h[n(r'),k1 ,ZJ E(r')e-ik 1", (16)

where now n(r t) becomes the unit normal vector to ý(r'), the undulating surface. The Sovietshave used a similar approach (15).

Substituting Eqs. (15) and (16) in Eq. (9) and specializing for beckscatter, k2 -*Iand n2 = -nl, the fdllowing approximation is derived:

Ef zK nj X f[n1 + Zan(r')] X [n(r') X h(r')] (r')e- 2 ik 1'rds'. (17)

The integration in Eq. (17) can be performed in the following sequence: fint, integrateover each slightly rough surface "patch" and then add the contributions from the "patches"

EB- J {In, X [n, + Z,n(p)j X [n(p) X h(p)]e- 2kjz(p(p)

f t(r)e- 2i(khx xh'1y*Y' d(patch) (18)

(p)

II

I

G.R. VALENZUELA AND M.B. LAING

PIS OZ....I {nj X [n, +Z~n(p)] X [n(p) X h(p)] nLjn X [n, + Z~n(k)J X [n(Q) X h(2)]j

F F • (r')•*(r")I(p) (R ) -

e-2iF Fkl'(P,'-klX( D)x +kly'(p )Y'-k ly )Y] d(p)d(D),(i

where x'v are in patch(p) and . ,y are in patch (9).

Neglecting the cros "'-rms (the contribution from two different patches), basically a Isumes that the fields f"'om diffetent patches are independent and the backscatter power willbe given by

z{-S0C In, X [n, + Zn(p)] X [n(p) X h(p)] 12.

(p)

ff t(r')t*(r' + Ar')e- [1Ix'Ax'+kly'AY'l d(Ax')d(Ay')}I(p)

(20)

Thus, for an incident plane wave when the slightly rough-ocal boundary condition is used,the backscattered power from a rough surface is given by Eq. (19). If the contributions fromthe various "patches" are mutually independent, the backscattered power may be approxi-mated by Eq. (20), which can be given the same physical interpretation as the compositesurface-cattering model formulation as proposed by Wright (6). This last result will be thebasis for the statistical model of sea clutter to Y developed in the following section.

Here it is appropriate to indicate that this analysis applies for an incident plane wave. Inpractical applications the incident electromagnetic radiation originates from an aperture offinite dimensions, and the incident electromagnetic radiation is a superposition of manyplane waves. Accordingly, the backscattered power should include an additional summationover the spectrum of the incident plane waves.

STATISTICAL MODEL OF SEA CLUTTER

As shown previously, the scatterLng results for certain types of rough surfaces, inparticular for the sea, can be simplified considerably by the assumptions of the com- Iposite surface-scattering model. It is recalled that the assumption of incoherent addi- 3tion has been used (power adds) in contrast with coherent addition in which the fields fromthe various slightuy rough "patches" add. S• I

I

NRL REPORT 7349 7

The backscattered power from each "patch" is the product of two factors: one factoris purely a function of the electrical properties of the surface and the local geometry, andanother factor is a function of the energy of the small-scale roughness t(r'); refer toEq. (20). Thus, if the illuminated area is of approximately one "patch" (i.e., narrow pulseand narrow antenna beam case), sea-clutter statistics may be modeled by a product of tworandom variables:

Z = XY, (21)

where X is a random variable dependent on the slopes of the large-scale roughness, T and Yis another random variable dependent on the statistics of the small-scale roughness •. Thevariables X and Y here are taken to be mutually independent.

Accordingly, the probability density functions (pdfs) of the random variables X and Yshould be of the forms

p(X) = P(I') I dX(22)

and

p(Y) = e-Y, (23)

where the mean value of Z has been included in X.

The reason for selecting these two pdfs should become more obvious as we proceed.Equation (22) follows directly from the law of transformation of pdfs, and Eq. (23) is thepdf of the energy spectra density of the small-scale roughness, which is assumed to beGaussian in amplitude. Therefore, the pdf of the random variable Z, according to elementaryprobability theory, should be given by (16)

pMZ = ( 1p(X)p Y Z (4

10

For the case involving illumination of several "patches" at one time (wide-pulse andwide-antenna beams), sea-clutter statistics may be represented by the sum random variable.

Zt = Z 1 + Z2 + + Zn. (25)

Clearly, the conditional density p( V/Ho) used in calculating the probability of falsealarm in Eq. (2) is Eq. (24) for the narrow-pulse and narrow-antenna beam or the pdf of therandom variable in Eq. (25) for the wide case.

A detailed development follows for the simpler case of single "patch" illumination.The random variable X in Eq. (21) can readily be identified with the normalized radar crosssection obtained for slightly rough scattering theory (7), thus (to first order), for horizontalpolarization


X(Oi)H 4w*k4 cot 4 Oja2 cos2 6T7L. + sin2 6[e(1 + ctj2) - ca] T11 112.

W(2k•c,2ky sin 8), (26)

and (to first order) for vertical polarization

X(Oi)V = 47rk 4 cot 4 O.a2 cois 2 B[e(i + aP)-C4 ]T11 11 + sin2 6T,112 IW(2ka,2ky sin 6), (27)

where

7C = Cos 5i f cos (0 + 0) cos 6, 0i = (1 - y7)11 2 , asin (0 + l),? = (1 -a ) 112,

e-1 e-1

j % (7+,e)21 11 I (-t +(er P

e is the complex dielectric constant of the surface, W(KZ, Ky) i the two~dimensiorul spec-

trum of the small-scale rougbness of the "patch" (i.e.,

f =(1/4) f foo W(K ,Ky)dMdKy),-00

which is taken to be constant on each "patch," and 0 and 6 are the "tt" angles with respectto the horizontal plane, parallel and perpendicular to the plane of hcklumce, respectively.

To have a unique relationship between X and slope r', side tilts cannot be included inthis formulation. Thus, in this statistical model, only tilts parallel to the plane of incidenceare to be included. Letting 6 = 0 in Eqs. (26) and (27), gives

_(jH=4 O40 (C 1) W(2k sin 0j, 0) (28)X(Oa)H = 4irk4 cos4 a.i (c i+/e -1 • 2 e)

1(cvd Oj + Vr-: .sn ,)2 W2 a ) (8

and

4k 4 (e- 1)[e(1 + sin2•0i) - sing 2 B 2

X(9i)v cos4 0, (e cos 8i + , sin2 9,)2 W(2k san 0O, 0). (29)

The slopes of the undulating surface are Gaussian distributed,

p(W) - I e-G• ')2/232, (3O)

and since tan ' = -', the pdf of X can be found from Eq. (22) using Eq. (30) and Eqs. (28)

or (29). Let W(KX, Ky) - 6 • i-3Kr-4 and initially take the cae of a pertectly conducting

sea, (where, e l -lao). Then, Eqs. (28) and (29) become

NRL REPORT 7349 9

X(Oi)H C C cot 4 0, (31)

and

X(Oi,)V C (cot2 0j + 2)2, (32)

where C - (3f/2) • 10-3.

After some straightforward algebra, for a perfectly conducting sea the pdcf for therandom variables X are

1 (A-tanr"\ 2

P(XH a T(33)and4C•

A3(1 +A tan r)2

andI

1 (B-tanr \2

sec2 " e 2 F+÷Btanr)/P(Xv) ' 4CSV(- B(B2 +2)(1+Btanr)2 ' (34)

wherer = 900 - 0 (grazing angle), A = (XHIC)1/4, and B - [(Xv/C)1/2 - 2] 1/2. The con-stants of proportionality in Eqs. (33) and (34) are obtained from the normalization of thepdf.

In Figures 2 through 5, Eq. (33) has been plotted for various grazing angles and rmsslopes of the undulating surface. As observed the pdf of XH is nearly Gaussian for small Sand tends to the exponential distribution for large S. For small grazing angles the tail of thepdf increases with large 8, a characteristic of the lognormal distribution; so seemingly wehave found that the lognormal properties of sea clutter may be explained in terms of thetilting of th? slightly rough patches by the large-scale roughness.

The probability distribution corresponding to Eq. (33) is

P( >x Ef otT Ian*IN (35)

where the upper sign applies for tan 0 and the lower sign applies for tan < 0, tan "A - tan T/1 + A tan rand

Eqatitt (35) has been plotted in Fig. 6 fr r- 404 and for sevrm vas of $. Figure' pums sme curis plotted acmnmnal probability pspe, wbere lokq nal distribuions plot


900 j

RMS SLOPE

7000 0.3'50.250.30

600 0

t""•U RMS SLOPEP'

0.05

2DO. 020

0 OCom 0002 0003 0004 0005 0006 0007 0.006 (009 0010

'H

Fig. 2-Denfty function of XH for pawfectly condheting san d 40-

as straight lines. Theas curvm indicate that the distribution of XH has lopwuwl character-istics, which is most evident for small S, and for large S the tail of the distributionwith increasing S.

The pdf and the distribution function for XV for a perfectly conducting m are nottoo interesting becauBe Eq. (32) approaches a constant toward the grazing incidence. T7I,to obtain realistic results for the distributions of XV the more exact express Eq. (29),which applies for a sea surface of finite conductivity, must be used. Te pdf mad theprobability distribution for XHf will not change drastically for the case of finite, but Iagconductivity.

From the functional dependence of Eqs. (28) and (29) on the ange of incklence andtilt angle, the inverse of those expressions can be approximated by the series

Nt.n(900 - ej) . aeXtI/N, (36)

Q*- 1

NRL REPORT 7349 1

900

8OO

700 RMS SLOPE

0.05

600 0.10

U 065z

o00 0,20

0•25

2O

0 3002 zI4 00 \ 0o3 0

200N

O2

0.05 010 0. 15 0.20 0.25 030 0.35

0.0511

0 k0 0.001 0002 0003 0004 0005 0006 0007 000 0009 0.000

XH

Fig. 3-Density function of XH for perfectly conducting m and 30-degre graing angi

where N 8 or 10 and the 's are constants. Equations (22) and (36) can be used to obtaiz.the general expression for the pdf of X:

N/_ 9aRX(R-N)IN

sec2 r R-Ip(X) 0C 8 S N

(1+ tan r aRQ.X1N)2

exp -•-•tani4 , (37)

and the probability distribution correponding to Eq. (37) is

P(X>X)4{Ef toot Erf( It 01) (38)21 VýFs) V Y'S


900

800

.'00 I600

~500

>.400 ,

IDI'tRMS SLOPE4 0.30Ao0.35 0.25 0120 015

100 .010

.05

0 0001 0002 0003 0.004 0005 0.006 0007 0006 0009 0OD0

XH

Fig. 4-Density function of XH for perfectly conducting - and 20-degm grazing angle

where again the upper sign applies for tan tO > 0 and the lower sign applies for tan 0' <0.In this case

NE aRXQ/N - tan r

tan 2n--

1 + tan r 4a2X N,

and the factor of proportionality in Eqs. (37) and (38), as before, is obtained from thenormalization condition P(X > 0) - 1.

A more careful investigation of Eqs. (37) and (38) show that the distribution of Xshould tend toward the Gaussian distribution when the last few coefficients dominate (thisis the case for vertical polarization). On the other hand if the first few coefficients dominatein Eq. (36), the distribution of X should tend toward the lognormal distribution (this is thecase for horizontal polarization. Thus, this model predicts that sea clutter for horizontalpolarization should tend toward the lognormal distibution.

NRL REPORT 7349 13

900

0 0

S0 oo

50014002

, 0.2 5

•02

200 0(;.5000. 03 00 0

XN

Fig. 5--Density functon of XH for perfectly conductin wo and 10-

Of course, the statistics of sea clutter in final analysis is related to the distribution ofthe random variable Z which also depends on the distribution of the other random variableY, which in our model is always an exponential distribution. Thus, any lognormal char-acteristics of sea-clutter distribution must come from the distribution of the random variableX. In general closed-form expressions cannot be obtained for the pdf and distribution func-tion of Z.

However, some limiting properties can be derived for the distribution of Z. For example,take the case of a glassy sea (no undulating function is present, S - 0). Clearly the pdf ofX will be a delta function:

p(X) 6 6(X -) (39)

and

p(Z) -w•-z/x, (40)

where is the mean value of X and also of Z, and Z is exponentially distributed for this cue.

14 G.R. VALENZUHLA AND M.B. LAING

107

05

rIcUY conducting nas and 40-degre paging04 MOOe

-50 -200-0

N ' 4~0 15 Z A4I

4 000

5 0-

~ 3 F~. 7Dinrlbtlo fuctin o XHforp rsof-l odutn n 0-erepzn

I 'hI\, To-nlinnra roaiiy.

I 10 -

* 50'

40-

30-~~~~ -Ditrbuio fucto of XHfr

7 N '~03D

ws 05k 0r3 020

0O21005!

-W -45 -40 36 *31 A -; -eb -O -5

X, t1t(bKLS)

NRL REPORT 7349 15

For increasing value of S (rougher seas), p(X) should develop a longer tail (see Figs.2-5); thus, p(Z) will also have a longer tail, which should make the distribution of Z tendtoward the lognormal distribution (in particular for horizontal polarization).

The moments of Eqs. (33), (34), and (37) do not exist because we have used a K-4

spectrum which has a singularity at K - 0. This in principle can be corrected by cutting offthe spectrum before K - 0. A shadowing function may also be included in Eq. (21) buthas not been attempted here.

IDENTIFICATION OF THE STATISTICS OF SEA CLUTFER OBTAINEDWITH THE NRL-4FR RIYUM

Quantitative analysis of sea clutter statistics is necessary for further development andimprovement of the statistical model of sea clutter. Thus, analysis has been performed onsear-cluttr .ata taken wit-i tae 4FR nynLem which generally uses pulse widths between 0.1and 0.5 psecond and whose antenna beamwidths are greater than 5 degrees. Accordinglythe illuminate0 area in this case contains many "patches" and no direct comparisons will bepossible with the predictions of the 3tatistical model, but any statistical information obtainedfor sea clutter will be helpful in assessing or updating the statistical model.

A detailed description of the 4FR system has already been given by Guinard (8); thus,here we should only indicate that the samples of sea clutter used in this investigation aresamples of backscattered power (or radar cross section ot the sea) which have been collectedby a logarithmic receiver with a dynamic range of over 45 decibels. The output of the re-ceiver is digitalized by means of a 30-nanosecond gate to seven-bit accuracy. The sea condi-tions and radar p-ameters of the measurements are summarized in Table 1. The experimentin the North Atlantic during February 1969 was performed in the neighborhood of OceanStations India and Juliet.

The Distribution Function

The samples of sea clut ter, for depression angles of 5 to 30 degrees, taken with the4FR system have been distributed and checked with the Kolmogorov-Smirnov test (17) ifthey come from an exponential population (Rayleigh envelope) or a lognormal population.In fitting the exponential distribution, the mean value of the exponential distribution wasadjusted for a minimum-maximum deviation between the sample and the population dis-tribution. In fitting the lognormnal distribution, the variance and the median values weretaken to be those of the sample.

The number cf independent samples of ma cautter was estimated by means of a stand-ard run-test (18) in an application of tho Wald-Wolfowitz test (19). The procedure usedhere is identical to that described by Schmidt (20) except that in our investigation sets of1024 samples are used instead of 2048. The ran-test, using the 95% level of significance,shows that 1 out of 4 samples is independent for X band, 1 out of 8 or 9 for C band, I outof 35 for L band, and I out of 120 to 130 is independent for P band. Of course, thesew aaverage numbers; in the analysis the exact number of independent samples was used for each

-- .-


Table IRadar and Sea Conditions

Dae'Wnd -Indicated Re.Pulse Wit

Date Wind Speed Wave Height Location Altitude Air Speed Rep. Pulse Width(m/sec) (M) (M) (m/eec) (pt M

2/11/69 24.5 8.0 OWS "1" 480 97 683 0.52/13/69 18 7.0 OWS "J" 460 103 603 0.52/14/69 20 7.5 OWS "J" 460 103 603 0.52/17/69 2.5 4.6 OWS 'T' 410 103 603 0.5 ,2/18/69 11 3.0 OWS "J" 460 103 603 0.51/23/70 6.2 3.7 Bermuda 180 103-115 683 0.51!26/70 7.5 1.5 Bermuda 750 89-108 683 0.51/27/70 1 8.2 1.8 Bermuda 180 87-108 683 0.51/27/70 8.2 1.8 Bermuda 610 87-110 683 0.5

cumulative distribution and in general the number of independent samples is proportional to

the radar frequency. No specific trends were found with polarization; the motion of the 3

aircraft is expected to decorrelate the samples of sea clutter faster. Thus, for sea cluttertaken with a stationary radar the number of independent samples should be smaller.

The Kolmogorov-Smirnov test, either at the 99% or at the 80% level of significance,shows that for calm seas and small sample sizes the cumulative distributions of sea clutterare acceptable as exponential or lognormal. However, for large sample sizes (data up to 30

seconds in this case) the maximum deviation between the distributions becomes larger thanthe acceptable lin,lit and should be rejected as belonging to the exponential or the log-normal family, with the smaller maximum deviations obtained against the exponential dis-tribution in most cases.

The maximum deviations for P- and L-band data were in general smaller than the maxi-mum deviations for C- and X-band data when compared with the exponential and the log-normal distributions. A trend occurs with polarization which may be more obvious in termsof the central moments, which are investigated in the next section.

In Figs. 8a-8x, some typical maximum deviations illustrate the above conclusions. In ithese figures, only the comparison with the exponential distribution is shown. The maximumdeviations in comparison with the lognormal distribution are not too different than thoseshown, except that, in general, they tend to be a little greater.

The outcome of the Kolmogorov-Smirnov test, although conclusive, is not very sati-fying sgnce the statisti"• of sea clutter are still not known.

NRL REPORT 7349 17

The Central Moments

"'he Kolmogorov-Smirnov test has demonstrated that sea clutter taken with the 4FRsystem is not .ponential nor lognormal distributed. To obtain a quantitative estimate ofthese svatistics, the first Live central moments of sea clutter should be computed from thedata, in decibels, to • vid additional errors in scaling of the data.

The central moments pl, A2, ..., un of a random variable which is exponentially or log-normal distributed are well known and can be used for comparison. The central momentsfor the logarithm of a random variable which is exponentially distributed are related to thepoligamma function (21)

[in r(x+1)1, (41)dAv

(where i'(x + 1) ia thule gamma runction). The central moments are:

A2 = 01 (0)- 1.64493

93 = Vi (0) =- 2.40411 (

94 = 01:i (0)= 6.49393 (

"1u5 = IV (0j = -24.88627

and 0(0) = -0.5772157 is related to the differm.nc hetwe. the natural logarithm of themean value of the exponential distribution and the mean kalue of the natitral logarithm ofthe random variable. The numerical values of the poligamma function hare been obtainedfrom Davis (22).

The central moments of the logarithm of a random variable which has a lognormal dis-tribution are those of the Gaussian distribution (14)

An - 1.3.5...(n - 1)•02n (43)

where n ;p 2 and is even. If n is odd, ;A, -f 0.

In Figs. 9 and 10 the put and the distribution function of an exponential and a log-normal random variable have been sketched for comparison.

Figures 11 through 20 illustrate the (median, mean) difference and the fint five

central moments of sea clutter (in decibels) taken with the NRL-4FR system for moderateand rough ses ccnditions. These figures also show the corresponding valuhs for the exponen-tufl and lognormal distribution. As observed the central moments of as clutter are in mort

cases intermediate between those of the exponentuw (Rayleigh envelope) and those of thelognormal distnbutijn, except for the fourth central moment.

18 C.R. VAiENZUELA AND M.B. LAING I

K Rttd65

Igo-f(a) Vertical poiarizatimn of P band on Feb. 11,

i "IIM.AEER OF INDEPENDENT SAMPLES

(b) Vertical polarization of L band on Feb. 11, "196b"

(c) Vertical poNrvtion of C band on FWb 11,1969

MAW" OF .CipthatuT SA'eLE

NRL REPORT 7349 19

(d) Vertical polarization of X band on Feb. 11, SD, -

1969

WMAER OF K)EPEM2EN SAWeLIS

ARh 165

II

11, 1969

(f) Horizontal polarization of L band on Feb. Feb

11, 1969 ••MUFN (W~r~ NWAAWIISWL

Fig. o-n•imuon deviation of cumulativ distrbtio of s clutt, taken with the 4FR s , f theexpoFwntial d eitributoio (Rayoif- u uneWopkk with the 99% and 80% aevel of s ylPeafnce from the

Kolmogorov-Smirnov test

...................................---.


(g) Horizontal polar.zation of C band on Feb.'9%

11, 1969

eLl

(h) Horizontal polarization of X band on Feb.11. 1969 .9%

80%

NLA4.IER OF trPrtD~ENT SUAULS

S(i) Vertical polarization of P band on Jan. 26,

NM OF MIPENDNT SALES

NRL REPORT 7349 21

" ~RUN 146

Vertical polarizablon of L band on Jan. 26, r

99%10 -,(k) Vertical polarization of C bend on Jan. 26,

@ 90 "1970

NLNUBER OF INDEPENrENT SAMPLES

9D%

(1) Vertical polarization of X bend on Jan. 26,1970

g

iii

Ic 4

WRIER OF INPNPTENT SAMPLES

Fig. 8-Maximum deviation of cumulative distribution of sea clutter, taken with the 4FR system, from theexponential distribution (Rayleigh envelope), with the 99% auJ 80% levels of saiificance from theKolmogorov-Smirnov test

I


II

(m) Horizontal polarization of P band on Jan.

, *26, 1970

-2

NMBER OF PEPE•nENT SAMPLES

RUN 146

(n) Horizontal polarization of L band on Jan. RU 4

id 0NUBE POPEDEJ~r SAMPLES

(o) Horizontal polarization of C band on Jan.26,61970

NMB~ER OF WicPENDNT SAMPLES 1

NRL REPORT 7349 23

RUN46

(p) Horizontal polarization of X band on Jan.26, 1970 9D%

Ib

ow

ei_ RUN 147 FBRO OEDNSAPS

i •- -(q) Vertical ýiolarization of P band on Jan. 26,

V3 1970

NUMBER_ OF INPENDENT SAMPLES 7

999%

(r) Vertical pol( rozation of L band on Jan. 26,1970 OFUME OF~tN SAWI.ES &V M 47

Ij •

Fig. 8-Maximum deviation of cumulative diatribution of oan clutter, taken with the 4PR systen, from theexponential distribution (Rayleigh envelope), with the 99% and 80% levels of significance from theKolmogorov-Smirnov tet

24 G.R. VALENZUELA AND M.I. LAING

99%

00% (a) Vertical polarization of C band on Jan. 26,1970

KUWR OF tCPENDENT SAMPLES

Ii(t) Vertical polarization of X bend on Jan. 26,

1970

MAW" OF WEPUNT" SAMPLES

6I4t (u) Horizontal polarization of P band on Jan.Is 26,19707E

idiNd OF f

17 sgPI~Mr1~.E

I

NRL REPORT 7349 25

(v) Horizontal polarization of L bend on Jan. l

26, 1970

14• 7 NLIME OF •MEPOW SA.rLES

v 'I "• (w) HorizonW4t polarization of C ,tund on Janl.'I

_ _ _ _ _ _ _ _ _ _ _ _d m c"

MAW OFt~ OCPNDN S.R RPtUM SA'

(xw) Horizontal polarization of C band on Jan.It 1 2. 19,707

MAEERWK OF MXEP(PC(N S'LES

Fig. 8-Mz) imum devistion of cumulative dit9ibution of 7cl0utter, taken with the 4FR system, from theexpoFenti di8tribution ocayi aigh envelope) withruto e l% and ith th e of siaysiten. from the

Kolmogorov-Smirno, tet


O0.MEDIAN EXPONENTIAL

0.b - (RAYLEIGH ENVELOPE)

0.06 ,'S007 LOGNORMAL

(WITH RAYLEIGH

/I • VARJANCE)S0.06 ' 1/

005-

S004-

,_.,.1 • I I I I '4L "-..L-25 -2C -15 -10 -5 0 5 to IS 20

(DECIBELS)

Fig. 9--Comparison of the exponential (Rayleigh envelope) andthe lognormal density functions

I0

, 0.905

W 4 LOC4OMAL0.4 (VARIANCE.30.9 d82

)

0.2 EXP0NINTLAL-~~0./

-25 -20 -1S -to -5 0 5 10 Is 20(DECMELS)

Fig. 10- Comparin of the exponential (Rayleigh envelope) andthe lognormal distribution functions

The central moments for vertical polarization are nearer to those fv, the exponentialdistribution, which agrees with the prediction of the model for the statistics of sea clutter.It wouid seem that the wide spread in the magritude of the centr-al moments =-y b an inu-i-cation that in general sea clutter is not stationary. Temporal variation of sea clutter has beeninvestigated by Schmidt (20).

With the central moments obtained for sea clutter, the true pdf and distribution func-tions should be reconstructable with no difficulty.

DISCUSSION OF RESULTS AND CONCLUSIONS

A model for the statistics of sea clutter has been developed using scattering theoryand the composite surface-scattering model, which uses the assumption that the rough

i0 5

NRL REPORT 7349 27

c -- EXPONENTIAL (RAYLEIGH ENVELOPE)

0 9 -. . . . . . . . . . . . . .-[ - - - i - - - - - - - - - - - - - - ---.

I08 -

w OS

w05-

0 04-~OAS03 0 O 0 5° (DEPRESSION ANGLE)

02 LOGNORAL 0 20ALEMEOIAN-MEAN 0 0

OJ

O I, I I_OO I 2 3 4 5 6 7 a 9 00

RADAR FREQUENCY (GIGAERT?)

Fig. 11--Medibn, mean difference of m dutter taken with the4FR system for modaate 'ls. Me wid ispeed wm 5 to 10metes/.coad. Solid points repmnt Vertical polarization andopen poiant, horizotal polarization.

os ,,| ----------- -

0I

0 -LOGNORALb•,2•,

o- MEDIA•N-I MCAN , 0 ý 30"

P ~ ~ ~ L, I I I i I I .. ,, t.-S; ; 3 4 5 6 7' 10 t

R.AWP FPKID49•NCY (GKA,•MEWz)

Fig. 12"-Median, mean difference of se chuem" taken with the4FR system for roulgh swas 7b wind speed was 15 to 20meteras/cond. Solid points represent vertical polurization andopen points, horizontal polariztion


40

56-

34 EXPONENTIAL (K-. LEIGH ENVELOPE)

32 .

0U 28 -a

w 26 El00rE24 -C

20 -- 01 5° (DEPRESSIONANGLE

0 2 3 4 5 7 7 6 9 10

RAWAR FREOUENCY (GIGAHERTZ)

Fig. 13-Variance of me clutter taken with the 4FFR sysiem formoderate mas. The wind speed was 5 to 10 meters/second. Thesolid points represent vertical polarization and the opel, points,horizontal polarization.

400

ý4 -- •EXPONENTIAL (RAYLEIGH ENVELOPE)

3---

2 0

>Z2- 0

20 I * (DEPRESS" A

F ig. 14--P na m c ol( t r tak in a w ith the 4 7R s yr tem fow

b'xiomtal p~ohe~im~ Is- 0

NRL R&IOPT 7349 29

0

- LOC.NORMAL OA5 0 5o (DEPRESSION~ AN4GLE)

-02- JO 10

-03- 0 y3O 0

-04-0

-05 o

a-of-

-12

-13

0 1 2 3 4 50 9 0

ft. 15--Thud anbnl wam t of sm c bkgu w~ the 4FR674m fcw r Im I mm Th wind mpd was 5 to 10 mtinuJ

se Te 'lfid -oa - aic m~Pahuaa and theOrra points, habmti pdmzfi3

rf:r (oE0P%[SSlON ANLE

-04-20-'3..--'

*0k p C a-

-2 3 4 13 4 8 10RAZAR!'WUIN1'CY tIAPý4- Z)

Fig. 1 e- Third cm tif of mackhe taktra with tho 4FR"sutera for ram Pm The vied qsped u 15 to 20 rmwtefr/

incoad The soid pett ,gmgw t vwtic pciatruston and th*ope points hcworhmW al zai


48

46- 0 0 51

S0 0 10*0 20-

42- 3

40-

38-0 I

± 36

34-- LOGNORMAL32m ISA / 0

30 ----------------------------- ----------------

28-EXPONENTIAL (RA--EIGH ENVELOPE)

26 /

o 2 3 4 5 6 7 8 9 10RAODR FREQUENCY (GIGAHERTZ)

Fig. 17-Fourth c:ntral moment of sea clutter taken with the4FR system for moderate seas. The wind speed was 5 to 10meters/second. The iolid points represent vertical polarizationand the open points, horizontal polarization.

48

46- 0 51

44- k 0 20"42- Q 0*421 -* (

40-

38i

N 36 -

S34 - 032 LOGNORMAL

32 -0

0

28

EXPONENTiAL (RAYLEIG4 ENVELOPE)26-/

I 2 3 4 5 6 7 8 9 10RADAR FRIECENCY IGIGAHERTZ)

Fig. 18-Fourth central moment of sea dutte.r taken with #I4FF syxtem for rough seas" The wind speed was 15 to 20metas/second. The wlid points represent vertical polarizationand the open ouin•,, horizontal polarization.

NRL REPORT 7349 31

surface "locally" is slightly rough. For the sea the "local"-slightly rough assumption isquite a reasonable approximation at most conventional radar frequencies.

The case of single "patch" illumination is treated in detail, with the model predictingthe statistics of sea clutter to be exponential (Rayleigh envelope) for a glassy sea (no large-scale roughness present), and for increasing roughness the distribution should develop alonger tail similar to that for a lognormal distribution, in particular for horizontal polariza-tion. Obviously, a mechanism has been found that yields sea clutter with a distribution oflognormal character.

Beckmann (23) has shown that by using physical optics for a perfectly conductingrough surface the scattered fields should be Rayleigh distributed everywhere except near thespecular direction. and for very rough surftces the scattered fields should be Rayleigh dis-tributed even in the specular direction. Thus, the non-Rayleigh statistics are obviously dueto the scattering propertlc. of the slightly rough "patches" and the tilting of the "patches"by the large-scale roughness.

As future input for updating the model, the statistics of sea clutter taken with the 4FRsystem have been analyzed by the Kolmogorov-Smirnov test and by a computation of thecentral moments of the distribution. The results indicate that in general sea clutter is notexponentially (Rayleigh envelope) nor lognormally distributed, and these distributions mayonly be the limiting distributions of sea clutter. The large spread of the central momentamay be an indication that sea clutter is not stationary; therefore, the optimum radar detec-tor must be of an adaptive nature.

Some other important results obtained in the empirical identification of sea clutter are:

1. For large sample sizes (greater than about 1000 independent samples) sea clutteris not exponential nor lognormal,

2. For small sample sizes (less than 200 independent samples) sea-clutter statisticsmay be approximated by both the exponential and the lognormal distributions,

3. Sea clutter for vertical polarization, in general, is more exponential than sea clutterfor horizonal polarizaticn,

4. Sea clutter for P and L band. in general, is more exponential than sea clutter for Cand X band,

5. The number of independent samples is roughly proportional to radar frequency,

6. Sea clutter for calm seas is more exponential than sea clutter for tougher seas.

In conclusion, it is possible ýo say that the model developed for the statistics of !ieaclutter and the empirical identification of data taken with the 4FR system indicate that thedistribution of sea clutter is intermediate between the exponential (RAyleigh envelope) andthe 1cgnormal distribution. Accordingly, the expected probability of false alarm for radars

I/


C

LOGNORMAL ;L5 0

-4 00

-4-

EXPONENTIAL (RAYLEIGH ENVELOPE)

4 ---- -

-9- 0 U

-10- 0 5* (DEPRESSION ANGLE)

-II "-0

0 10-

-12 {-•S30,20,

% P Il I • I L 1 X!

I 2 3 4 5 6 7 a 9 tORADAR FREQUENCY (GIGA•ERTZ)

Fig. 19--Fifth metzal nmmt of ma clutter taken with the 4FRsystm= for moderate s•ea. The wind speed was 5 to 10 meters/open points, horizontal polarization.

170- 2K

-4--

EXPONENTIAL (RAYLEIGH ENVELOPE)

7 - -- - - - - - - - - ---- - -

=. -0 _

-9 --

-II r0 5* (DEPRESSION ANGLE)

-12 GLONORMAL L5 0 )230,

0 1 2 3 4 5 6 7 a 9 10RADAR FREQUENCY (GIGAHERTZ)

Fig. 20-Fifth central moment of Rea clutter, taken with the 4FR

system, for rough seas. The wind speed was 15 to 20 meters/second. The solid points represent vertical polarization and theopen points, horizontal polarization.

I

NRL REPORT 7349 33

operating in the sea environment should be larger than that predicted for an exponentiallydistributed clutter, in particular for horizontal polarization and toward grazing incidence.

ACKNOWLEDGMENTS

The authors thank N. W. Guinard for suggesting the application of the compositesurface-scattering model to investigating the statistics of sea clutter.

REFERENCES

1. D. Middleton, An Introduction to Statistical Communication Theory, McGraw-Hill,New York, 1960.

-. J.I. Marcum, "A Statistical Theory of Target Detection by Pulsed Radar," Rand Corp.,Santa Monica, Calif., RM-754, Dec. 1, 1947, and RM-753, July 1, 1948.

3. P. Swerling, "Probability of Detection for Fluctuating Targets," Rand Corp., SantaMonica, Calif., RM-1217, Mar. 17, 1954, and IRE Trans. Iniorm. Theory, IT-6, 269-308 (1960).

4. HoL. Van Trees, Detection, Estimation, and Modulation Theory, Wiley, New York,1968.

5. G.V. Trunk and S.F. George. "Detection of Targets. in Non-C-xa,,_ian. Sea Clutter,"IEEE Trans., AES-6, No. 5, 620-628 (1970).

6. J.W. Wright, "A New Model for Sea Clutter," IEEE Irns. AP-16, No. 2, 217-223(1968).

7. G.R. Valenzuela, "Scattering of Electromagnetic Waves from a Tilted Slightly RoughSurface," Radio Sci. 3, No. 11, 1057-1066 (1968).

8. N.W. Guinard, "The NRL Four-Frequency Radar System," Report of NRL Progress,May 1969, pp. 1-10.

9. N.W. Guinard and J.C. Daley, "An Experimental Study of a Sea Clutter Model," Proc.IEEE 58, No. 4, 543-550 (1970).

10. N.W. Guinard, J.T. Ranscae, Jr., and J.C. Daley, "The Variation of the NRCS of theSea with Increasing Roughness," J. Geophys. Rae. 76, No. 6, 1525-1538 (1971).

11. G.R. Valenzuela, M.B. Laing, and J.C. Daley, "Ocean Spectra for the High FrequencyWaves as Determined from Airborne Radar Measurements," J. Marine Re& 29, No. 2,69-84 (1971).

II


12. G.R. Valenzuela and M.B. Laing, "Study of Doppler Spectra of Radar Sea Echo," J.Geophys. Res. 75, No. 3, 551-563 (1970).

13. A.W. Maue, "Zur Formulierung eines allgemeinen Beugungsproblemns durch eineInt'gralgleichung," Z. Physik 126, 601-618 (1949).

14. K.M. Mitzner, "An Integral Equation Approach to Scattering from a Body of FiniteConductivity," Radio Sci. 2, No. 12, 1459-1470 (1967).

15. F.G. Bass, I.M. Fuks, A.I. Kalmykov, I.E. Ostrovsky, and A.D. Rosenbezg, "Very HighFrequency Radiowave Scattering by a Distrubed Ser Surface," Trans. IEE•E AP-16,No. 5, 554-568 (1968).

16. W.B. Davenport, Jr., and W.L. Root, Chapter 3 in An Introdu,-tion to the " heory of jRandom Signals and Noise, McGraw-Hill, New York, 1958.

17. F.J. Massey, Jr,, "The Kolmogorov-Smirnov Test for Goodness of Fit," J. Am. Stat.Assoc. 46, 68-78 (1951).

18. J.S. Bendat and A.G. Piersol, Measurement aid Analysis of Random Data, Wiley, NewYork, 1966.

19. A. Wald and J. Wolfowitz, "On a Test Whether Two Samples are from the Same Popula-tion," Ann. Math. Stat. 11, 1.-7-162 (1940).

20. KR. Schmidt, "Statistical Time-Varying and Distribution Properties of High-ResolutionRadar Sea Echo," NRL Report 7150, Nov. 9, 1970.

21. M.M. Siddigui, "Statistical Inference for Rayleigh Distributions," Radio Science(Journal of Research NBS/USNC-URSI) 68D, No. 9, 1005-1010 (1968).

22. H.T. Davis, Tables of the Mathematical Functions, Vol. II, The Principia Press ofTrinity University, San Antonio, Texas, 1935.

23. P. Beckmann anid A. Spizzichino, Chapter 1 The Scattering of Electromagnetic Wavesfrom Rough Surfaces, New York, Pergamon Press, 1963.

NRL Report 7349 - DTIC · 2018-11-08 · NRL Report 7349 A310310B/652A/A R02101-002 c" (formerly A310310B/652A/1 elf AOAOI lC.E)A1 R008-01-020) OlI~t~1lV~l ~ 9b. OTHER thie repot,)REPORT

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